Eighteenth Order Convergent Method for Solving Non-Linear Equations

In this paper, we suggest and discuss an iterative method for solving nonlinear equations of the type f(x) = 0 having eighteenth order convergence. This new technique based on Newton’s method and extrapolated Newton’s method. This method is compared with the existing ones through some numerical examples to exhibit its superiority. AMS Subject Classification: 41A25, 65K05, 65H05.

Which is same as Halley's method having third order convergence which requires three functional evaluations.The efficiency index of this method is 3 3 1.4422 = .
A three step Predictor-corrector Newton's Halley method (PCNH) suggested by Mohammed and Hafiz 10 , is given by: For a given x 0 , compute x n+1 by using ...(1.7) This method has eighteenth order convergence and its efficiency index is 8 18 1.4352 The three step Eighteenth Order Extrapolated Newton's method (EOCM) developed by Vatti et.al., 12 is given by: For a given x 0 , compute x n+1 by the iterative schemes This method has eighteenth order convergence and its efficiency index is 8 18 1.4352 = .
In section 2, we consider the Second derivative free Eighteenth Order Extrapolated Newton's method and discuss the convergence criteria of this method in section 3. Few numerical examples are considered to show the superiority of this method in the concluding section.

Second Derivative Free Eighteenth Order Convergent Method (Seocm)
Considering the Eighteenth Order Extrapolated Newton's method (EOCM) (1.8) with (1.9) and (1. .Therefore, the algorithm (2.1) has eighteenth order convergence and its efficiency index is which is same as that of the method (1.7) and (1.8).

introduction=
It is well known that a wide class of problems, which arises in diverse disciplines of mathematical and engineering science can be studied by the nonlinear equation of the form f (x) = 0 ...(1.1)Where : fD R R ⊂→ is a scalar function on an open interval D and f (x) may be algebraic, transcendental or combined of both.The most widely used algorithm for solving (1.1) by the use of value of the function and its derivative is the well known quadratic convergent Newton's method (NM) given by ...(1.2) starting with an initial guess x 0 which is in the vicinity of the exact root x* .The efficiency index of Newton's method is 2 2 1.4142 = .The Extrapolated Newton's method (ENM) suggested by V.B.Kumar, Vatti et.al., 11 which is developed by extrapolating Newton's method (1.2) introducing a parameter ' α n ' given by VATTI et al., Orient.J. Comp.Sci.& Technol., Vol.10(4) 829-835 (2017) ...(1.3)Here the optimal choice for the parameter ' α n ' is ...(1.4)Where ...(1.5)Combining (1.3), (1.4) and (1.5), one can have ...(1.6)